Applications of Complexity Analysis in Clinical Heart Failure

Abstract

Heart failure is known to influence heart rhythm in patients. Complexity analysis techniques, including techniques associated with entropy, have great potential for providing a better understanding of cardiac rhythms, and for helping research in this area. We review the analysis principles of conventional time-domain analysis, frequency-domain analysis and of newer complexity analysis. We then illustrate the techniques using real clinical data, allowing a comparison of the techniques, and also of the differences between normal heart rate variability and that associated with heart failure.

A. Appendix

1.1 A.1 RR Sequence Normalized Histogram

The general construction procedure for the RR sequence normalized histogram is summarized as follows [53]:

Given an RR sequence {RR ₁, RR ₂,…, RR _N }, where N denotes the sequence length. The maximum (RR _max) and minimum (RR _min) values were firstly determined to calculate the range in the sequence:

$$ {RR}_{\text{range}}={RR}_{\max }-{RR}_{\min } $$

(11.5)

The threshold a = 0.1 × RR _range is set and then the left-step parameter H _l and right-step parameter H _r can be calculated as

$$ {H}_l=\frac{RR_{\text{mean}}-\left({RR}_{\min }+\alpha \right)}{5} $$

(11.6)

$$ {H}_r=\frac{\left({RR}_{\max }-\alpha \right)-{RR}_{\text{mean}}}{5} $$

(11.7)

where RR _mean denotes the mean value of the RR sequence. Based on these parameters, the RR _i is divided into seven sections. Table 11.1 details the element division rules.

Table 11.1 Element division rules for constructing the RR sequence normalized histogram

The element percentage p _i in each of the sections is calculated as follows:

$$ {p}_i=\frac{P_i}{N}\begin{array}{cc}\hfill \hfill & \hfill i=1,2,\cdots, 7\hfill \end{array} $$

(11.8)

In a rectangular coordinate system, the p _i corresponding to the seven sections (i.e. L1, L2, L3, C, R3, R2 and R1) is drawn to form the normalized histogram. Three quantitative indices can be obtained from the normalized histogram. They are respectively named as center-edge ratio (CER), cumulative energy (CE) and range information entropy (RIEn) and are defined as [53]:

$$ \text{CER}=\frac{p_4}{p_1+{p}_2+{p}_6+{p}_7} $$

(11.9)

$$ \text{CE}=\sum_{i=1}^7{p}_i^2 $$

(11.10)

$$ \text{RIEn}=-{\sum}_{i=1}^7{p}_i\ast \ln {p}_i $$

(11.11)

1.2 A.2 Sample Entropy (SampEn)

The algorithm for SampEn is summarized as follows [29]: For the HRV series x(i), 1 ≤ i ≤ N, forms N − m + 1 vectors ${X}_i^m=\left\{x(i),x\left(i+1\right),\cdots, x\left(i+m-1\right)\right\}$, 1 ≤ i ≤ N − m + 1. The distance between two vectors ${X}_i^m$ and ${Y}_i^m$ is defined as: ${d}_{i,j}^m=\underset{k=0}{\overset{m-1}{ \max }}\left|x\left(i+k\right)-x\left(j+k\right)\right|$. Denote ${B}_i^m(r)$ the average number of j that meets ${d}_{i,j}^m\le r$ for all 1 ≤ j ≤ N − m, and similarly define ${A}_i^m(r)$ by ${d}_{i,j}^{m+1}$. SampEn is then defined by:

$$ \text{SampEn}\left(m,r,N\right)=- \ln \left(\sum_{i=1}^{N-m}{A}_i^m(r)/\sum_{i=1}^{N-m}{B}_i^m(r)\right) $$

(11.12)

wherein the embedding dimension is usually set at m = 2 and the threshold at r = 0.2 × sd (sd indicates the standard deviation of the HRV series under-analyzed) [57, 60].

1.3 A.3 Fuzzy Measure Entropy (FuzzyMEn)

The calculation process of FuzzyMEn is summarized as follows [33, 56]:

For the RR or PTT segment x(i) (1 ≤ i ≤ N), firstly form the local vector sequences ${XL}_i^m$ and global vector sequences ${XG}_i^m$ respectively:

$$ \begin{array}{c} \hfill{\quad } {XL}_i^m=\left\{x(i),x\left(i+1\right),\cdots, x\left(i+m-1\right)\right\}-\overline{x}(i)\hfill \\[8pt] {XG}_i^m\!=\left\{x(i),x\left(i+1\right),\cdots, x\left(i+m-1\right)\right\}-\overline{x}\hfill \end{array}\kern1em 1\le i\le N-m $$

(11.13)

The vector ${XL}_i^m$ represents m consecutive x(i) values but removing the local baseline $\overline{x}(i)$, which is defined as:

$$ \overline{x}(i)=\frac{1}{m}\sum_{k=0}^{m-1}x\left(i+k\right)\kern1em 1\le i\le N-m $$

(11.14)

The vector ${XG}_i^m$ also represents m consecutive x(i) values but removing the global mean value $\overline{x}$ of the segment x(i), which is defined as:

$$ \overline{x}=\frac{1}{N}\sum_{i=1}^Nx(i) $$

(11.15)

Subsequently, the distance between the local vector sequences ${XL}_i^m$ and ${XL}_j^m$, and the distance between the global vector sequences ${XG}_i^m$ and ${XG}_j^m$ are computed respectively as:

$$ \begin{array}{l}{dL}_{i,j}^m=d\left[{XL}_i^m,{XL}_j^m\right]=\underset{k=0}{\overset{m-1}{ \max }}\left|\left(x\left(i+k\right)-\overline{x}(i)\right)-\left(x\left(j+k\right)-\overline{x}(j)\right)\right|\\[8pt] {}{dG}_{i,j}^m=d\left[{XG}_i^m,{XG}_j^m\right]=\underset{k=0}{\overset{m-1}{ \max }}\left|\left(x\left(i+k\right)-\overline{x}\right)-\left(x\left(j+k\right)-\overline{x}\right)\right|\end{array} $$

(11.16)

Given the parameters n _L , r _L , n _G and r _G , we calculate the similarity degree ${DL}_{i,j}^m\left({n}_L,{r}_L\right)$ between the local vectors ${XL}_i^m$ and ${XL}_j^m$ by the fuzzy function $\mu L\left({dL}_{i,j}^m,{n}_L,{r}_L\right)$, as well as the similarity degree ${DG}_{i,j}^m\left({n}_G,{r}_G\right)$ between the global vectors ${XG}_i^m$ and ${XG}_j^m$ by the fuzzy function $\mu G\left({dG}_{i,j}^m,{n}_G,{r}_G\right)$:

$$ \begin{array}{l}{DL}_{i,j}^m\left({n}_L,{r}_L\right)=\mu L\left({dL}_{i,j}^m,{n}_L,{r}_L\right)= \exp \left(-\frac{{\left({dL}_{i,j}^m\right)}^{n_L}}{r_L}\right)\\[8pt] {}{DG}_{i,j}^m\left({n}_G,{r}_G\right)=\mu G\left({dG}_{i,j}^m,{n}_G,{r}_G\right)= \exp \left(-\frac{{\left({dG}_{i,j}^m\right)}^{n_G}}{r_G}\right)\end{array} $$

(11.17)

We define the functions ϕL ^m(n _L , r _L ) and ϕG ^m(n _G , r _G ) as:

$$ \begin{array}{l}\phi {L}^m\left({n}_L,{r}_L\right)=\frac{1}{N-m}\sum_{i=1}^{N-m}\left(\frac{1}{N-m}\sum_{j=1}^{N-m}{DL}_{i,j}^m\left({n}_L,{r}_L\right)\right)\\[10pt] {}\phi {G}^m\left({n}_G,{r}_G\right)=\frac{1}{N-m}\sum_{i=1}^{N-m}\left(\frac{1}{N-m}\sum_{j=1}^{N-m}{DG}_{i,j}^m\left({n}_G,{r}_G\right)\right)\end{array} $$

(11.18)

Similarly, we define the function ϕL ^m + 1(n _L , r _L ) for m + 1 dimensional vectors ${XL}_i^{m+1}$ and ${XL}_j^{m+1}$ the function ϕG ^m + 1(n _G , r _G ) for m + 1 dimensional vectors ${XG}_i^{m+1}$ and ${YG}_j^{m+1}$:

$$ \begin{array}{l}\phi {L}^{m+1}\left({n}_L,{r}_L\right)=\frac{1}{N-m}\sum_{i=1}^{N-m}\left(\frac{1}{N-m}\sum_{j=1}^{N-m}{DL}_{i,j}^{m+1}\left({n}_L,{r}_L\right)\right)\\[10pt] {}\phi {G}^{m+1}\left({n}_L,{r}_L\right)=\frac{1}{N-m}\sum_{i=1}^{N-m}\left(\frac{1}{N-m}\sum_{j=1}^{N-m}{DG}_{i,j}^{m+1}\left({n}_G,{r}_G\right)\right)\end{array} $$

(11.19)

Then the fuzzy local measure entropy (FuzzyLMEn) and fuzzy global measure entropy (FuzzyGMEn) are computed as:

$$ \begin{array}{l}\text{FuzzyLMEn}\left(m,{n}_L,{r}_L,N\right)=- \ln \left(\phi {L}^{m+1}\left({n}_L,{r}_L\right)/\phi {L}^m\Big({n}_L,{r}_L\Big)\right)\\[8pt] {}\text{FuzzyGMEn}\left(m,{n}_G,{r}_G,N\right)=- \ln \left(\phi {G}^{m+1}\left({n}_G,{r}_G\right)/\phi {G}^m\Big({n}_G,{r}_G\Big)\right)\end{array} $$

(11.20)

Finally, the FuzzyMEn of RR segment x(i) is calculated as follows:

$$ \begin{aligned}[b] &\text{FuzzyMEn}\left(m,{n}_L,{r}_L,{n}_G,{r}_G,N\right)=\text{FuzzyLMEn}\left(m,{n}_L,{r}_L,N\right)\\[4pt] &\quad+\text{FuzzyGMEn}\left(m,{n}_G,{r}_G,N\right)\end{aligned} $$

(11.21)

In this study, the local similarity weight was set to n _L = 3 and the global vector similarity weight was set to n _G = 2. The local tolerance threshold r _L was set equal to the global threshold r _G, i.e., r _L = r _G = r. Hence, the formula (11.21) becomes:

$$ \text{FuzzyMEn}\left(m,r,N\right)=\text{FuzzyLMEn}\left(m,r,N\right)+\text{FuzzyGMEn}\left(m,r,N\right) $$

(11.22)

For both SampEn and FuzzyMEn, the entropy results were only based on the three parameters: the embedding dimension m, the tolerance threshold r and the RR segment length N.

1.4 A.4 Lempel-Ziv (LZ) Complexity

The calculation process of LZ complexity is summarized as follows [21, 59]:

• 1. For a binary symbolic sequence R = {s ₁, s ₂,…, s _n }, let S and Q denote two strings, respectively, and SQ is the concatenation of S and Q, while the string SQπ is derived from SQ after its last character is deleted (π means the operation to delete the last character in the string). v(SQπ) denotes the vocabulary of all different substrings of SQπ. Initially, c(n) = 1, S = s ₁, Q = s ₂, and so SQπ = s ₁;

• 2. In summary, S = s ₁ s ₂, …, s _r , Q = s _r+1, and so SQπ = s ₁ s ₂, …, s _r ; if Q belongs to v(SQπ), then s _r+1, that is, Q is a substring of SQπ, and so S does not change, and Q is updated to be s _r+1 s _r+2, and then judge if Q belongs to v(SQπ) or not. Repeat this process until Q does not belong to v(SQπ);

• 3. Now, Q = s _r+1 s _r+2, …, s _r+i , which is not a substring of SQπ = s ₁ s ₂, …, s _r s _r+1,…, s _r+i-1, so increase c(n) by one;

• 4. Thereafter, S is renewed to be S = s ₁ s ₂, …, s _r+i , and Q = s _r+i+1;

• 5. Then the procedures repeat until Q is the last character. At this time c(n) is the number of different substrings contained in R. For practical application, c(n) should be normalized. It has been proved that the upper bound of c(n) is

$$ c(n)<\frac{n}{\left(1-{\varepsilon}_n\right){ \log}_{\alpha }(n)}, $$

(11.23)

where ε _n is a small quantity and ε _n → 0 (n → ∞). In fact,

$$ \underset{n\to \infty }{ \lim }c(n)=b(n)=\frac{n}{{ \log}_{\alpha }(n)}. $$

(11.24)

• 6. Finally, LZ complexity is defined as the normalized output of c(n):

$$ C(n)=\frac{c(n)}{b(n)}, $$

(11.25)

where C(n) is the normalized LZ complexity, and denotes the arising rate of new patterns within the sequence.